# Summing over Geometries in String Theory

Research paper by **Lorenz Eberhardt**

Indexed on: **25 Feb '21**Published on: **24 Feb '21**Published in: **arXiv - High Energy Physics - Theory**

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#### Abstract

We examine the question how string theory achieves a sum over bulk geometries
with fixed asymptotic boundary conditions. We discuss this problem with the
help of the tensionless string on $\mathcal{M}_3 \times \mathrm{S}^3 \times
\mathbb{T}^4$ (with one unit of NS-NS flux) that was recently understood to be
dual to the symmetric orbifold $\text{Sym}^N(\mathbb{T}^4)$. We strengthen the
analysis of arXiv:2008.07533 and show that the perturbative string partition
function around a fixed bulk background already includes a sum over
semi-classical geometries and large stringy corrections can be interpreted as
various semi-classical geometries. We argue in particular that the string
partition function on a Euclidean wormhole geometry factorizes completely into
factors associated to the two boundaries of spacetime. Central to this is the
remarkable property of the moduli space integral of string theory to localize
on covering spaces of the conformal boundary of $\mathcal{M}_3$. We also
emphasize the fact that string perturbation theory computes the grand canonical
partition function of the family of theories
$\bigoplus_N\text{Sym}^N(\mathbb{T}^4)$. The boundary partition function is
naturally expressed as a sum over winding worldsheets, each of which we
interpret as a `stringy geometry'. We argue that the semi-classical bulk
geometry can be understood as a condensate of such stringy geometries. We also
briefly discuss the effect of ensemble averaging over the Narain moduli space
of $\mathbb{T}^4$ and of deforming away from the orbifold by the marginal
deformation.